algebra 1b chapter 10 section 1 factors. algebra standard 11.0 – students apply basic factoring...
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Algebra 1B
Chapter 10 Section 1
Factors
Algebra Standard
• 11.0 – Students apply basic factoring techniques to second and simple third degree polynomials . These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
Prime and Composite Numbers
• Some numbers have 2 factors, the number itself and 1. This is called a prime number.
• Whole numbers that have more than two factors, such as 12, are called composite numbers.
Example of Prime and Composite
Prime Numbers less than
20.
Composite numbers
less than 20
Neither Prime nor Composite
2,3,5,7,11,13,17,19
4,6,8,9,10,12,14,15,16,1
8
0,1
Example 1
Directions: Find the factors of each number. Then state each math problem as prime or composite number.
72To find the factors of 72, list all pairs of whole numbers whose product is 72.
1 X 72, 2 X 36, 3 X 24, 4 X 18, 6 X 12, 8 X 9
• The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
• Since 72 has more than two factors, it is a composite factor.
Your Turn
• Find the factor for 25. Then state each math problem as a prime or a composite number.
1 X 25
5 X 5
The factors of 25 are 1, 5, 25. Since there are more than 2 numbers it is a composite.
Example 2
Directions: Find the factors of each number. Then state each math problem as a prime or a composite. 37
• There is only one pair of whole numbers whose product is 37.
• 1 X 37• The factors of 37 are 1 and 37.
Therefore, 37 is a prime number.
Your Turn
Directions: Find the factor of 23, and state as a prime or a composite.
1 X 23
Since there are only two factors, 23 is a prime number.
Directions: Find the factors of 79 and state each number as a prime or a composite.
79
1 X 79
Since 1 and 79 only have two factors, 79 is a prime number.
Directions: Find the factors of 51, and determine if it is a composite number or a prime number.
51
1 X 51
3 X 17
1, 3, 17, and 51 are factors of 51. Since there are more than two factors in the number 51 it is a composite number.
Example 3
Directions: Factor each monomial.
12a2b
4 · 3 · a · a · b What number still needs to be factored?
2 · 2 · 3 · a · a · b
Factor 100mn3
100mn3
50 · 2 · m · n · n · n
25 · 2 · 2 · m · n · n · n
5 · 5 · 2 · 2 · m · n · n · n
Factor 25x2
25x2
5 · 5 · x · x
5 · 5 · x · x
Your Turn
Factor each monomial.
15ab2
3 · 5 · a · b · b
Factor
84xz2
42 · 2 · x · z · z
21 · 2 · 2 · x · z · z
7 · 3 · 2 · 2 · x · z · z
Factor
⁻36b3
6 · 6 · b · b · b
2 · 3 · 2 · 3 · b · b · b
2 · 2 · 3 · 3 · b · b · b
Two or more numbers may have some common prime factors.
Example:36 and 42
36 = 2·2·3·3 Line up the common factors
42 = 2 · 3 · 7
The integers 36 and 42 have 2 and 3 as common prime factors.
The product, which means multiplication, of these prime factors, 2 · 3 or 6.
This is called the GREATEST COMMON FACTOR (GCF) of 36 and 42.
Definition of GCF
Greatest common factor- The greatest common factor of two or more integers is the product of the prime factors common to the integers.
Your Definition
What is the definition of Greatest Common Factor (GCF) in your own words?
Examples of GCF
Directions: Find the GCF of each set of numbers or monomials.
24, 60, 7224 = 2·2·2·3 Find the prime factorization of each
number.
60 = 2·2· 3·5 Line up as many common factors as possible.
72 = 2·2·2·3·3 Circle the common factors.
The GCF of 24, 60, and 72 is 2·2·3 or 12.
Find the GCF of 15 and 8.15 = 3 · 58 = 2 · 2 · 2
There are no common prime factors. The only common factor is 1. So, the GCF of 15 and 8 is 1.
Find the GCF of 15a2b and 18ab.
15a2b = 3 · 5 · a · a · b18ab = 2 · 3 · 3 · a · b
The GCF of 15a2b and 18ab is 3 · a · b or 3ab.
Your TurnFind the GCF or each set of numbers or monomials.
75, 100, and 5075 = 3 · 5 · 5100 = 2 · 2 · 5 · 550 = 2 · 5 · 5
The GCF of 75, 100. and 150 is 5 · 5 or 25.
What is the GCF of 5a and 8b?
5a = 5 · a8b = 8 · b
What is the GCF when there are no common factors?
Since there are no terms that are equal the GCF is 1.
What is the GCF of 24ab2c and 60a2bc?
24ab2c = 2 · 2 · 2 · 3 · a · b · b · c60a2bc = 5 · 2 · 2 · 3 · a · a · b · c
The GCF is 2 · 2 · 3 · a · b · c, which equals 12abc.
Algebra 1B
Chapter 10 Section 1
Factors